338 MATHEMATICS. 



regular treatise on the Differential Calculus, was the Analysis of Infi- 

 nites, published in France, in 1699, by the Marquis de L'Hopital ; 

 and the first elementary treatise upon it in England, was published by 

 Hayes, in 1704. Brook Taylor's theorem, published in 1715, and 

 Maclaurin's theorem, deduced from Taylor's, have become the basis 

 of the calculus of finite differences, or increments and series, on 

 which Lagrange has founded his whole theory of the Calculus. 

 Maclaurin, in his Treatise of Fluxions, published in 1742, first sub- 

 jected the principles of this science to strictly geometrical proof; and 

 they were demonstrated analytically by Lagrange, in 1772. 



James Bernouilli, a friend of Leibnitz, distinguished himself by 

 the application of the Calculus to the elastic spring, the logarithmic 

 spiral, and to the most difficult isoperimetrical curves : and his 

 brother John Bernouilli, though doubtless his inferior, did much 

 to promote this science ; particularly in his examination of expo- 

 nential functions, about the year 1697; and in his application of 

 Leibnitz's method of differencing from curve to curve. The import- 

 ant method of partial differences, first applied by Euler, was 

 extended by D'Alembert, in studying the vibrations of a musical 

 string ; and still farther extended by Euler himself, in his Investiga- 

 tio Functionum, published in 1762. La Grange, in 1760, invented 

 the Calculus of Variations ; which Euler was one of the first to 

 adopt, and which La Place has successfully applied to the planetary 

 perturbations. The developement of functions in series, has been 

 facilitated by the labors of La Place ; as also by Hindenburg's combi- 

 natory analysis, and by Arbogast's calculus of derivations, invented 

 in the year 1800. Of farther improvements, by Clairaut, Fontaine, 

 Legendre, Cousin, and others, we have no room here to speak. 



Our further notice of this science will be very brief, and com- 

 prised under the two divisions of the Differential, and the Integral 

 Calculus. 



1. The immediate object of the Differential Calculus, is, having 

 given the relation of two quantities, or fluents, to find the ratio of 

 their differentials, or fluxions. The name of differentials is given 

 to the increments of quantities, when supposed to become infinitely 

 small or zero ; but though the increments themselves thus disappear, 

 their ratio or proportion to each other does not disappear, but becomes 

 an exact and definite mathematical quantity, having important appli- 

 cations. In this branch of mathematics, quantities are considered as 

 either constant, or variable; the former being expressed by the first, 

 and the latter by the last letters of the alphabet. When the value 

 of one quantity depends upon that of another, the former is said to 

 be a function of the latter. Thus, in the equation, y = ax + b,yis 

 said to be an explicit function of x ; and x is said to be an implicit 

 or implied function of y but a and b represent constant quantities. 

 In this example, if x increases in value, y is also increased ; and 

 hence y may be called an increasing function of x. Functions are 

 also distinguished as either algebraic or transcendental; the latter 

 >emg either logarithmic, or circular functions, which cannot be ex- 

 pressed by a limited number of algebraic terms, but only by a series ; 

 as i/ = log. x, or y = sin a-. 



